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The Pythagorean identity is a fundamental principle in trigonometry that states the sum of the squares of sine and cosine of an angle equals one. In mathematical terms, it is expressed as:

• \( \sin^2 x + \cos^2 x = 1 \)

This identity is very useful for simplifying expressions involving these functions.

In the context of the given function, \( f(x) = \sin^2 x + \cos^2 x \), applying the Pythagorean identity simplifies the entire expression to just \( f(x) = 1 \). This transformation reduces the complexity and helps in further calculations or differentiations, as seen in the steps of the solution.

A constant function is one where the function value remains the same, regardless of the input value. In other words, it does not change as the variable changes, and it can be represented as:

• \( f(x) = c \), where \( c \) is a constant.

These functions are represented graphically by horizontal lines in the Cartesian plane.

For our exercise, simplified \( f(x) = 1 \) is a perfect example of a constant function. Since the value never changes, irrespective of the value of \( x \), it allows us to apply a specific rule of differentiation known for constants, simplifying our task further.

Differentiation is an operation in calculus used to find the rate at which a function changes. The basic rules of differentiation help us easily compute derivatives without having to resort to first principles each time.

One crucial rule is for constant functions. The derivative of a constant function \( f(x) = c \) is always zero because a constant does not change with respect to any variable.

Applying this rule to the function from our exercise, \( f(x) = 1 \), we find that its derivative, \( f'(x) \), is zero. This comes from the rule that the change in a constant value is simply none, hence the derivative vanishes.

These rules are not only applicable to constant functions but extend to more complex functions involving powers, products, and quotients, thereby forming the backbone of calculus.